Families of solutions to the KPI equation and the structure of their rational representations of order N (original) (raw)

Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation

Universal Journal of Mathematics and Applications

Multi-parametric solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants are constructed in function of exponentials. A representation of these solutions as a quotient of wronskians of order 2N2N2N in terms of trigonometric functions is deduced. All these solutions depend on 2N−12N-12N−1 real parameters. A third representation in terms of a quotient of two real polynomials depending on 2N−22N-22N−2 real parameters is given; the numerator is a polynomial of degree 2N(N+1)−22N(N+1)-22N(N+1)−2 in xxx, yyy and ttt and the denominator is a polynomial of degree 2N(N+1)2N(N+1)2N(N+1) in xxx, yyy and ttt. The maximum absolute value is equal to 2(2N+1)2−22(2N+1)^{2}-22(2N+1)2−2. We explicitly construct the expressions for the first third orders and we study the patterns of their absolute value in the plane (x,y)(x,y)(x,y) and their evolution according to time and parameters.\\ It is relevant to emphasize that all these families of solutions are real and non singular.

Fredholm representations of solutions to the KPI equation, their wronkian versions and rogue waves

2016

We construct solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants. We deduce solutions written as a quotient of wronskians of order 2N. These solutions called solutions of order N depend on 2N − 1 parameters. When one of these parameters tends to zero, we obtain N order rational solutions expressed as a quotient of two polynomials of degree 2N (N + 1) in x, y and t depending on 2N − 2 parameters. So we get with this method an infinite hierarchy of solutions to the KPI equation.

Multiparametric Families of Solutions of the Kadomtsev–Petviashvili-I Equation, the Structure of Their Rational Representations, and Multi-Rogue Waves

Theoretical and Mathematical Physics, 2018

We construct solutions of the Kadomtsev-Petviashvili-I equation in terms of Fredholm determinants. We deduce solutions written as a quotient of Wronskians of order 2N. These solutions, called solutions of order N , depend on 2N −1 parameters. They can also be written as a quotient of two polynomials of degree 2N (N + 1) in x, y, and t depending on 2N −2 parameters. The maximum of the modulus of these solutions at order N is equal to 2(2N + 1) 2. We explicitly construct the expressions up to the order six and study the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters.

Other Families of Rational Solutions to the KPI Equation

Asian Research Journal of Mathematics

Aims / Objectives: We present rational solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of polynomials in x, y and t depending on several real parameters. We get an infinite hierarchy of rational solutions written as a quotient of a polynomial of degree 2N(N + 1) - 2 in x, y and t by a polynomial of degree 2N(N + 1) in x, y and t, depending on 2N - 2 real parameters for each positive integer N. Place and Duration of Study: Institut de math´ematiques de Bourgogne, Universit´e de Bourgogne Franche-Cont´e between January 2020 and January 2021. Conclusion: We construct explicit expressions of the solutions in the simplest cases N = 1 and N = 2 and we study the patterns of their modulus in the (x; y) plane for different values of time t and parameters. In particular, in the study of these solutions, we see the appearance not yet observed of three pairs of two peaks in the case of order 2.

Multiparametric Rational Solutions of Order N to the KPI Equation and the Explicit Case of Order 3

Archives of Current Research International

We present multiparametric rational solutions to the Kadomtsev-Petviashvili equation (KPI). These solutions of order N depend on 2N − 2 real parameters. Explicit expressions of the solutions at order 3 are given. They can be expressed as a quotient of a polynomial of degree 2N(N +1)−2 in x, y and t by a polynomial of degree 2N(N +1) in x, y and t, depending on 2N − 2 real parameters. We study the patterns of their modulus in the (x,y) plane for different values of time t and parameters.

Other patterns for the first and second order rational solutions to the KPI equation

HAL (Le Centre pour la Communication Scientifique Directe), 2022

We present rational solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of polynomials in x, y and t depending on several real parameters. We get an infinite hierarchy of rational solutions written as a quotient of a polynomial of degree 2N (N + 1) − 2 in x, y and t by a polynomial of degree 2N (N + 1) in x, y and t, depending on 2N − 2 real parameters for each positive integer N. We construct explicit expressions of the solutions in the simplest cases N = 1 and N = 2 and we study the patterns of their modulus in the (x, y) plane for different values of time t and parameters. In particular, in the study of these solutions, we see the appearance not yet observed of three pairs of two peaks in the case of order 2.

From particular polynomials to rational solutions to the KPI equation

HAL (Le Centre pour la Communication Scientifique Directe), 2022

We construct solutions to the Kadomtsev-Petviashvili equation (KPI) from particular polynomials. We obtain rational solutions written as a second derivative with respect to the variable x of a logarithm of a determinant of order n. So we get with this method an infinite hierarchy of rational solutions to the KPI equation. We give explicitly the expressions of these solutions for the first five orders.

Rational solutions to the KPI equation of order 7 depending on 12 parameters

2018

We construct in this paper, rational solutions as a quotient of two determinants of order 2N = 14 and we obtain what we call solutions of order N = 7 to the Kadomtsev-Petviashvili equation (KPI) as a quotient of 2 polynomials of degree 112 in x, y and t depending on 12 parameters. The maximum of modulus of these solutions at order 7 is equal to 2(2N + 1) 2 = 450. We make the study of the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6. When all these parameters grow, triangle and ring structures are obtained.

Fredholm and Wronskian representations of solutions to the KPI equation and multi-rogue waves

Journal of Mathematical Physics, 2016

We construct solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants. We deduce solutions written as a quotient of Wronskians of order 2N. These solutions, called solutions of order N, depend on 2N − 1 parameters. When one of these parameters tends to zero, we obtain N order rational solutions expressed as a quotient of two polynomials of degree 2N(N + 1) in x, y, and t depending on 2N − 2 parameters. So we get with this method an infinite hierarchy of solutions to the KPI equation.